Optimal. Leaf size=18 \[ i \sin (x)-\cos (x)-i \tanh ^{-1}(\sin (x)) \]
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Rubi [A] time = 0.10, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {3518, 3108, 3107, 2638, 2592, 321, 206} \[ i \sin (x)-\cos (x)-i \tanh ^{-1}(\sin (x)) \]
Antiderivative was successfully verified.
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Rule 206
Rule 321
Rule 2592
Rule 2638
Rule 3107
Rule 3108
Rule 3518
Rubi steps
\begin {align*} \int \frac {\sec (x)}{i+\cot (x)} \, dx &=-\int \frac {\tan (x)}{-\cos (x)-i \sin (x)} \, dx\\ &=i \int (-i \cos (x)-\sin (x)) \tan (x) \, dx\\ &=i \int (-i \sin (x)-\sin (x) \tan (x)) \, dx\\ &=-(i \int \sin (x) \tan (x) \, dx)+\int \sin (x) \, dx\\ &=-\cos (x)-i \operatorname {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\sin (x)\right )\\ &=-\cos (x)+i \sin (x)-i \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (x)\right )\\ &=-i \tanh ^{-1}(\sin (x))-\cos (x)+i \sin (x)\\ \end {align*}
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Mathematica [B] time = 0.04, size = 44, normalized size = 2.44 \[ -\cos (x)+i \left (\sin (x)+\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 33, normalized size = 1.83 \[ {\left (-i \, e^{\left (i \, x\right )} \log \left (e^{\left (i \, x\right )} + i\right ) + i \, e^{\left (i \, x\right )} \log \left (e^{\left (i \, x\right )} - i\right ) - 1\right )} e^{\left (-i \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.39, size = 29, normalized size = 1.61 \[ \frac {2 i}{\tan \left (\frac {1}{2} \, x\right ) - i} - i \, \log \left (\tan \left (\frac {1}{2} \, x\right ) + 1\right ) + i \, \log \left (\tan \left (\frac {1}{2} \, x\right ) - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.30, size = 34, normalized size = 1.89 \[ \frac {2 i}{\tan \left (\frac {x}{2}\right )-i}+i \ln \left (\tan \left (\frac {x}{2}\right )-1\right )-i \ln \left (\tan \left (\frac {x}{2}\right )+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.57, size = 45, normalized size = 2.50 \[ -\frac {2}{\frac {i \, \sin \relax (x)}{\cos \relax (x) + 1} + 1} - i \, \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 1\right ) + i \, \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 21, normalized size = 1.17 \[ -\mathrm {atanh}\left (\mathrm {tan}\left (\frac {x}{2}\right )\right )\,2{}\mathrm {i}+\frac {2{}\mathrm {i}}{\mathrm {tan}\left (\frac {x}{2}\right )-\mathrm {i}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec {\relax (x )}}{\cot {\relax (x )} + i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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